A very long-term transient event preceding the 2011 Tohoku earthquake

Geodetic transients have been observed in various subduction zones. The 2011 Tohoku earthquake occurred in one of the most active subduction zones globally, the Japan Trench subduction zone (JTSZ). However, no geodetic transient (except afterslip and so on) had been reported in the JTSZ before the Tohoku earthquake. Here we show that a large transient event, with duration longer than any reported previously, occurred in the JTSZ preceding the Tohoku earthquake. We calculate tectonic deformations at Global Positioning System stations along the JTSZ by removing the effects of nearby Mw 6–8 earthquakes. We identify temporal changes in these deformations, deriving 9-year deviation records from regular deformations due to slip deficit at the plate boundary. We perform an inversion of the deviations to obtain the source model of their root event. The relationship between the obtained transient event and Tohoku earthquake is shown through Coulomb stress change and seismic supercycle simulation. Slow slip events have been observed in different subduction zones, but their relationship to megathrust earthquakes remains elusive. Here, the authors postulate that a transient event may have led to the 2011 Tohoku earthquake as the hypocentre falls within a zone of positive Coloumb stress change.

G eodetic transients have been observed in various subduction zones [1][2][3] and have long been believed to be the key phenomena associated with megathrust earthquakes, although the nature of this relationship remains elusive 4 . Indeed, it has been reported that geodetic transients might have preceded megathrust earthquakes 5 . For example, long-term slow slip events (SSEs), which are those with duration longer than 10 7 s (ref. 3) were observed before the 1983 Japan Sea earthquake along the boundary of the Eurasian and North American plates 6 and before the 2012 Ometepec earthquake in the southern Mexico subduction zone 7 . However, in one of the most active subduction zones in the world, the Japan Trench subduction zone (JTSZ), where the 2011 Tohoku earthquake 8,9 occurred with a moment magnitude (M w ) of 9.0, no geodetic transient had been reported previously; rather, only afterslip, short-term silent earthquakes and recovery of plate coupling had been reported 3,10 . In fact, even in the period immediately before the Tohoku earthquake, only short-term transients around the foreshock were reported 11,12 . Nevertheless, tectonic deformations due to drag of the overriding plate coupled with the subducting plate (slip deficit) were being observed continuously in the JTSZ, and any geodetic transients that occurred in the region should have affected these tectonic deformations.
To investigate this phenomenon, we first analyse the temporal changes in tectonic deformations before the Tohoku earthquake. We use a data set from the dense Global Positioning System (GPS) network GEONET, operated by the Geospatial Information Authority of Japan (GSI). In previous studies [13][14][15][16] , the slip deficit rate in the JTSZ was imaged using partial data sets covering the period of seismic quiescence before the 2003 Tokachi-oki earthquake. In this study, we use the whole data set from 21 March 1996 to 8 March 2011 (one day before the foreshock).

Results
Time series of deformation. We converted the daily coordinates of the GEONET stations in the Tohoku and Kanto districts along the JTSZ to time series of horizontal deformation (see Methods). Sinusoidal annual variations and the co-seismic and post-seismic signals of nearby M w 6-8 earthquakes are apparent in the results of this conversion (Fig. 1a,b). In addition, stationary westward trends were obtained at all stations for the seismic quiescence period before 2002 (red lines in Fig. 1b), and we assumed that these represent regular deformations that were due to slip deficit, as in all previous studies [13][14][15][16] . We conducted an additional analysis of the time series; the results shown in Supplementary Note 1 and Supplementary Fig. 1 can also support our assumption.
We removed the annual variations and co-seismic and postseismic signals from the time series by fitting trigonometric, Heaviside and logarithmic functions. Then, we performed regression analyses for the pre-2002 parts of the time series to determine the trends of regular slip deficit deformations and removed the obtained trends from the time series, as for the Tokai SSE 1 and Bungo Channel SSE 17 . If the slip deficit deformations were the only parameters affecting the results, the data should be distributed around the zero line. However, the obtained data began to deviate from the zero line around 2002 (Fig. 1c). This deviation was eastward at the central and southern stations, where movement was accelerated at the time of the 2005 M w 7.2 Miyagi-oki earthquake, but westward at the northern stations,  Characteristics of the sources of deviations. Figure 2 shows the distribution of measured total deviations (pink arrows). As found in deformation accelerations 18 and deformations including postseismic signals 9,19,20 , large eastward deviations were observed along the eastern coast of central and southern Tohoku with a broad extension to the west. Meanwhile, westward deviations are obvious in the data obtained for northern Tohoku, although there was no significant deviation in the Kanto district. Small southward deviations were found extending along the western coast of northern Tohoku, as has been found previously in deformation accelerations 18 . This implies that the sources of the large eastward and westward deviations were located far from the western coast. Small southward deviations were also detected along the boundary between central and northern Tohoku, where the large eastward and westward deviations cancelled each other out.
On the basis of the above-mentioned evidence, we assumed the two sources to be located on deeper parts of the plate boundary beneath and off the eastern coast of central and southern Tohoku or northern Tohoku. The northern source for the westward deviations was interpreted previously to be a result of recovery of plate coupling 10 or temporary acceleration of subduction 21 . These interpretations cannot be applied to the southern source, because this source generated deviations in the opposite direction. However, the deviation direction and source position relative to the Tohoku earthquake are similar to those of the Tokai SSE 1 relative to the 1854 M 8.4 Tokai earthquake and the Bungo Channel SSE 17 Fig. 1c. A more similar case was found in the Alaska subduction zone, where long-term SSEs with duration of several years were observed downdip to the source region of the 1964 M 9.2 Alaska earthquake 22 . Therefore, we can consider the southern source to be a phenomenon like a longterm SSE, but this would have a total duration of about 9 years, longer than any long-term SSE reported previously; thus, we refer to it as a very long-term transient event, following the terminology in simulation studies 23,24 .
In contrast to short-term SSEs, long-term SSEs and their relatives are not always accompanied by tremor activities, although, if an episodic tremor and slip (ETS) zone exists in the vicinity of a long-term SSE, the SSE can excite tremor activity in the ETS zone 4 . However, no ETS zones have been found in the JTSZ, and the very long-term transient event was not accompanied by tremor activities.
Inversion of the deviations for finite source models. Because deformation due to slip deficit is westward and slip deficit is traditionally modelled according to a normal faulting mechanism consisting of backslips 25,26 , the very long-term transient event for eastward deviations can be modelled according to a reverse faulting mechanism consisting of forward slips. Consequently, we located a dipping rectangular reverse fault on the deeper part of the plate boundary in central and southern Tohoku to explore the source characteristics of the very long-term transient event by analysing the deviations. However, we also had to introduce a dipping rectangular normal fault in northern Tohoku, because this northern backslip source affected deviations not only along but also beyond the boundary between northern and central Tohoku.

Discussion
To examine the relationship between the very long-term transient event and the regular process consisting of the Tohoku earthquake and the slip deficit, we next computed changes in the Coulomb failure function (DCFF) 28 , as was conducted previously for the Tokai SSE 1 and simulated transient events 29 . The computations were performed for the focal mechanism and hypocentral depth of the Tohoku earthquake. A comparison of the DCFF distribution with the slip distribution of the Tohoku earthquake 8 in Fig. 3 indicates that the northern half zone of the large positive DCFF resulting from the very long-term transient event coincides with the main rupture area of the Tohoku earthquake, except for its trenchward extension. However, the northern backslip source distributed only negative or zero DCFF values throughout the rupture area of the Tohoku earthquake. Therefore, only the very long-term transient event affected the Tohoku earthquake. The DCFF values of the very long-term transient event, from near its upper edge to the hypocentre of the Tohoku earthquake, varied from þ 0.2 to þ 0.02 MPa, with an average of þ 0.1 MPa, around the mainshock hypocentre. This average is 20 times larger than the value found in the Tokai earthquake source area due to the Tokai SSE 1 , which did not lead to a megathrust earthquake. Finally, we conducted a numerical simulation of megathrust earthquake cycles (seismic supercycles 30 ) to examine the possibility of a very long-term transient event in the JTSZ. We used the two-dimensional mechanical model 31 in Supplementary  Fig. 2 and the method described in Methods. The velocity of the Pacific plate motion relative to the North American plate (V pl ) is assumed to be 8.5 cm per year off central Tohoku, according to a plate kinematic model 32 . As in the original simulation 31 , the result of the simulation in Supplementary Fig. 3 indicates that a megathrust earthquake (large step) occurs every several hundred years, with large slips in the shallower regions around the Tohoku earthquake hypocentre (A, B and C) and smaller slips in the deeper region (D). Under such conditions, slip deficit accumulates (represented by long slopes) and M 7-class earthquakes (small steps in the slopes) occur during a supercycle between megathrust earthquakes. Moreover, several very long-term transient events (flat parts in the slopes) also occur in D during a supercycle, with the last one occurring just before and leading to a megathrust earthquake.
In short, the interseismic period of a supercycle should consist primarily of three phenomena: the slip deficit of the regular process, M 7-class earthquakes and very long-term transient events. Therefore, the 2002 to March 2011 parts of the time series shown in Fig. 1b should represent the sum of the effects of these three phenomena. However, because the crustal and in-slab earthquakes (dark green stars in Fig. 1a) do not belong to the category of M 7-class earthquake defined above, we removed their co-seismic and post-seismic signals (dark green lines in Fig. 1b) from the time series. We measured composite deformations due to the three phenomena from the revised time series, as shown with pink arrows in Supplementary Fig. 4. Then, we performed an inversion of the composite deformations for a source model extending along the Japan Trench, and computed DCFF values due to the resultant slip distribution (purple contours in Supplementary Fig. 4), finding the composite DCFF around the mainshock hypocentre to be B þ 1 MPa.
Considering the results of both the seismic supercycle simulation and the two DCFF computations, it is proven that a very long-term transient event occurred over the 9 years preceding the 2011 Tohoku earthquake. We also suggest that this transient event could have led to the advancement of the occurrence time 33 of the Tohoku earthquake, contributing 10% to the composite DCFF. The above simulation did not perfectly reproduce these findings; performing an extensive set of simulations with new inputs would provide further understanding of the physics governing such transient events and their relationship to a megathrust earthquake like Tohoku.

Methods
Interseismic deformation. The GSI is analysing GPS data from their GEONET using the Bernese GPS software (version 5.0), IGS ephemerides, and ITRF2005 (ref. 34). The results of the analyses are being published as the F3 solutions for the daily coordinates of the GEONET stations 34  coordinate system VIII, IX or X (ref. 35). The horizontal components of deformation at each station were calculated from the obtained x-and y-coordinates.
Source inversion scheme. We performed the source inversion using a leastsquares method with constraints 36 , although the positivity constraint on slip vectors was not applied because the observed deviations include the effects of the very long-term transient event and the northern backslip source; thus, we had to model reverse-and normal-faulting sources simultaneously. Smoothness constraints with digital Laplacians remained, and their weights were determined by minimizing Akaike's Bayesian Information Criterion 37 . Green's functions were computed using the average one-dimensional velocity structure in and off the Tohoku and Kanto districts and the propagator matrix method with dynamic-static separation 38 .
Seismic supercycle simulation. The simulation was conducted using a twodimensional mechanical model 31 and the method for media under the rate-and state-dependent friction law 39 . The plate boundary shape in the original model was revised to fit to the actual shape in and off central Tohoku. We also modified the distributions of parameters in the rate-and state-dependent friction law (a À b, L and s n eff ) (ref. 39) along the plate boundary. The unstable zone of a-bB0 was extended to a depth over 60 km and the effective normal stress s n eff was decreased at depths deeper than 40 km ( Supplementary Fig. 2). Other minor changes in a-b, s n eff and L were also applied as shown in Supplementary Fig. 2.